ProductLog[-(Log[x]/x)] == -Log[x] /; 0 <= x <= E

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ProductLog", "[", RowBox[List["-", FractionBox[RowBox[List["Log", "[", "x", "]"]], "x"]]], "]"]], "\[Equal]", RowBox[List["-", RowBox[List["Log", "[", "x", "]"]]]]]], "/;", RowBox[List["0", "\[LessEqual]", "x", "\[LessEqual]", "\[ExponentialE]"]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mi> W </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mi> x </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mn> 0 </mn> <mo> &#8804; </mo> <mi> x </mi> <mo> &#8804; </mo> <mi> &#8519; </mi> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> W </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ln /> <ci> x </ci> </apply> <apply> <power /> <ci> x </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <ci> x </ci> </apply> </apply> </apply> <apply> <leq /> <cn type='integer'> 0 </cn> <ci> x </ci> <exponentiale /> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ProductLog", "[", RowBox[List["-", FractionBox[RowBox[List["Log", "[", "x_", "]"]], "x_"]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", RowBox[List["Log", "[", "x", "]"]]]], "/;", RowBox[List["0", "\[LessEqual]", "x", "\[LessEqual]", "\[ExponentialE]"]]]]]]]]

2001-10-29